On triple systems and strongly regular graphs

The block graph of a Steiner triple system of order v is a ( v ( v − 1 ) / 6 , 3 ( v − 3 ) / 2 , ( v + 3 ) / 2 , 9 ) strongly regular graph. For large v, every strongly regular graph with these parameters is the block graph of a Steiner triple system, but exceptions exist for small orders. An explanation for some of the exceptional graphs is here provided via the concept of switching. (Group divisible designs corresponding to) Latin squares are also treated in an analogous way. Many new strongly regular graphs are obtained by switching and by constructing graphs with prescribed automorphisms. In particular, new strongly regular graphs with the following parameters that do not come from Steiner triple systems or Latin squares are found: ( 49 , 18 , 7 , 6 ) , ( 57 , 24 , 11 , 9 ) , ( 64 , 21 , 8 , 6 ) , ( 70 , 27 , 12 , 9 ) , ( 81 , 24 , 9 , 6 ) , and ( 100 , 27 , 10 , 6 ) .